- Open Access
Some Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums
© Zheng and Fu; licensee Springer 2012
- Received: 3 August 2012
- Accepted: 14 December 2012
- Published: 28 December 2012
Some new generalized Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums are established in this paper. To illustrate the validity of the established inequalities, we present some applications for them, in which new explicit bounds for the solutions of certain infinite sum-difference equations are deduced.
- nonlinear discrete inequalities
- Volterra-Fredholm type inequalities
- sum-difference equations
In recent years, many researchers have focused on various generalizations of the known Gronwall-Bellman inequality [1, 2], which provide explicit bounds for unknown solutions of certain difference equations, and a lot of such generalized inequalities have been established in the literature [3–20] including the known Ou-Iang inequality . In , Ma generalized the discrete version of Ou-Iang’s inequality in two variables to a Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of the solutions of certain Volterra-Fredholm type difference equations. But since then, few results on Volterra-Fredholm type discrete inequalities have been established. Recent results in this direction include the works of Zheng , Ma , Zheng and Feng  to our best knowledge. We notice that the Volterra-Fredholm type discrete inequalities in [22–24] are constructed by an explicit function in the left-hand side (see [, Theorems 2.5, 2.6], [, Theorems 2.1, 2.5, 2.6, 2.7], [, Theorems 5, 8, 10, 11]).
Motivated by the works in [22–24], in this paper, we establish some new generalized Volterra-Fredholm type discrete inequalities involving four iterated infinite sums with the right-hand side denoted by an arbitrary function , which are of more general forms. To illustrate the usefulness of the established results, we also present some applications for them and study the boundedness of the solutions of certain Volterra-Fredholm type infinite sum-difference equations.
Throughout this paper, ℝ denotes the set of real numbers and , and ℤ denotes the set of integers, while denotes the set of nonnegative integers. In the next of this paper, let , where , and let be two constants. If U is a lattice, then we denote the set of all ℝ-valued functions on U by and denote the set of all -valued functions on U by . Finally, for a function , we have provided .
Lemma 2.1 [, Lemma 2.1]
Since is selected from Ω arbitrarily, then substituting with in (12), we get the desired inequality (3). □
where is defined in (16).
Combining (19), (22) and (23), we get the desired result. □
The proof for Corollary 2.4 can be completed by setting , , , in Theorem 2.3.
Combining (34), (36) and (37), we get the desired result. □
Combining (46), (48) and (49), we get the desired result. □
The proof for Theorem 2.7 is similar to the combination of Theorem 2.5 and Theorem 2.6, and we omit the details here.
Remark 2.8 We note that the inequalities established in Theorems 2.3, 2.5-2.7 are essentially different from the results in [22–24] as the left-hand side of the inequalities established here is an arbitrary function . Furthermore, if we set , , then Theorem 2.5 reduces to [, Theorem 2.5].
In this section, we present some applications for the results established above. Similar to the applications in [22–24], we research a certain Volterra-Fredholm sum-difference equation and derive some new bounds for its solutions.
where , is an odd number, , .
Combining (53)-(55), we can deduce the desired result. □
Combining (58)-(60), we can deduce the desired result. □
The authors thank the referees very much for their careful comments and valuable suggestions on this paper.
- Gronwall TH: Note on the derivatives with respect to a parameter of solutions of a system of differential equations. Ann. Math. 1919, 20: 292–296. 10.2307/1967124MathSciNetView ArticleMATHGoogle Scholar
- Bellman R: The stability of solutions of linear differential equations. Duke Math. J. 1943, 10: 643–647. 10.1215/S0012-7094-43-01059-2MathSciNetView ArticleMATHGoogle Scholar
- Ou-Iang L:The boundedness of solutions of linear differential equations . Shuxue Jinzhan 1957, 3: 409–418.Google Scholar
- Pachpatte BG: Inequalities for Differential and Integral Equations. Academic Press, New York; 1998.MATHGoogle Scholar
- Cheung WS: Some new nonlinear inequalities and applications to boundary value problems. Nonlinear Anal. TMA 2006, 64: 2112–2128. 10.1016/j.na.2005.08.009View ArticleMATHGoogle Scholar
- Zhao XQ, Zhao QX, Meng FW: On some new nonlinear discrete inequalities and their applications. J. Inequal. Pure Appl. Math. 2006., 7: Article ID 52Google Scholar
- Yang EH: On some nonlinear integral and discrete inequalities related to Ou-Iang’s inequality. Acta Math. Sin. Engl. Ser. 1998, 14: 353–360. 10.1007/BF02580438View ArticleMATHGoogle Scholar
- Cheung WS: Some discrete nonlinear inequalities and applications to boundary value problems for difference equations. J. Differ. Equ. Appl. 2004, 10: 213–223. 10.1080/10236190310001604238View ArticleMATHGoogle Scholar
- Meng FW, Ji DH: On some new nonlinear discrete inequalities and their applications. J. Comput. Appl. Math. 2007, 208: 425–433. 10.1016/j.cam.2006.10.024MathSciNetView ArticleMATHGoogle Scholar
- Ma QH, Pečarić J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 2008, 69: 393–407. 10.1016/j.na.2007.05.027MathSciNetView ArticleMATHGoogle Scholar
- Pachpatte BG: Inequalities applicable in the theory of finite differential equations. J. Math. Anal. Appl. 1998, 222: 438–459. 10.1006/jmaa.1998.5929MathSciNetView ArticleMATHGoogle Scholar
- Pachpatte BG: On some new inequalities related to a certain inequality arising in the theory of differential equations. J. Math. Anal. Appl. 2000, 251: 736–751. 10.1006/jmaa.2000.7044MathSciNetView ArticleMATHGoogle Scholar
- Cheung WS, Ma QH, Pečaric̀ J: Some discrete nonlinear inequalities and applications to difference equations. Acta Math. Sci., Ser. B 2008, 28: 417–430.View ArticleGoogle Scholar
- Deng SF: Nonlinear discrete inequalities with two variables and their applications. Appl. Math. Comput. 2010, 217: 2217–2225. 10.1016/j.amc.2010.07.022MathSciNetView ArticleMATHGoogle Scholar
- Jiang FC, Meng FW: Explicit bounds on some new nonlinear integral inequality with delay. J. Comput. Appl. Math. 2007, 205: 479–486. 10.1016/j.cam.2006.05.038MathSciNetView ArticleMATHGoogle Scholar
- Ma QH, Cheung WS: Some new nonlinear difference inequalities and their applications. J. Comput. Appl. Math. 2007, 202: 339–351. 10.1016/j.cam.2006.02.036MathSciNetView ArticleMATHGoogle Scholar
- Ma QH: N -independent-variable discrete inequalities of Gronwall-Ou-Iang type. Ann. Differ. Equ. 2000, 16: 813–820.Google Scholar
- Pang PYH, Agarwal RP: On an integral inequality and discrete analogue. J. Math. Anal. Appl. 1995, 194: 569–577. 10.1006/jmaa.1995.1318MathSciNetView ArticleMATHGoogle Scholar
- Pachpatte BG: On some fundamental integral inequalities and their discrete analogues. J. Inequal. Pure Appl. Math. 2001., 2: Article ID 15Google Scholar
- Meng FW, Li WN: On some new nonlinear discrete inequalities and their applications. J. Comput. Appl. Math. 2003, 158: 407–417. 10.1016/S0377-0427(03)00475-8MathSciNetView ArticleMATHGoogle Scholar
- Ma QH: Some new nonlinear Volterra-Fredholm-type discrete inequalities and their applications. J. Comput. Appl. Math. 2008, 216: 451–466. 10.1016/j.cam.2007.05.021MathSciNetView ArticleMATHGoogle Scholar
- Zheng B: Qualitative and quantitative analysis for solutions to a class of Volterra-Fredholm type difference equation. Adv. Differ. Equ. 2011., 2011: Article ID 30Google Scholar
- Ma QH: Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications. J. Comput. Appl. Math. 2010, 233: 2170–2180. 10.1016/j.cam.2009.10.002MathSciNetView ArticleMATHGoogle Scholar
- Zheng B, Feng QH: Some new Volterra-Fredholm-type discrete inequalities and their applications in the theory of difference equations. Abstr. Appl. Anal. 2011., 2011: Article ID 584951Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.